On Singular Interval-Valued Iteration Groups - Zdun, Marek C. (Frontiers in Applied Mathematics and Statistics, 2016)
Let I = (a, b) and L be a nowhere dense perfect set containing the ends of the interval I and let φ : I → ℝ be a non-increasing continuous surjection constant on the components of I L and the closures of these components be the maximal intervals of constancy of φ. The family Ft, t ∈ ℝ of the interval-valued functions Ft(x): = φ−1[t + φ(x)], x ∈ I forms a set-valued iteration group. We determine a maximal dense subgroup T ⊊ ℝ such that the set-valued subgroup Ft, t ∈ T has some regular properties. In particular, the mappings T ∍ t → Ft(x) for t ∈ T possess selections ft(x) ∈ Ft(x), which are disjoint group of continuous functions.